The duty per

Example (8.) A quarter of malt, duty-paid, sells for 70s. quarter is £1.085. How much per cent. does the duty enhance the price?

£3.5

Here, it is evidently necessary, as a first step, to deduct the duty from the selling price which includes it.

Example (9.) 1200 gallons of pure alcohol, weighing 7.938 lbs. per gallon, consist of 4969-9 parts by weight of carbon, 1242.5 of hydrogen, and the rest of oxygen. Required the per-centage proportion of oxygen ?

1200 X 7.938 9525.6 total weight.

=

4969-91242.5 6212.4

lbs. of oxygen. 3313.2

9525.6 : 100 :: 3313.2 : 34.78

Answer. 34.78 per cent.

This question is simply a case of division into proportional parts. (See p. 119.) Example (10.) By an investment of money, 5 per cent. was gained: by a second investment (including capital and profit), 7 per cent. was lost. There was then remaining, £900. What was the sum originally invested?

1st. £100 gained £5. Total £105

2nd. £100 lost £7, therefore £105 lost £7.35; for 100 105 :: 7 : 7:35 105-7.35 = £97.65 sum left out of each £100

97.65: 900 : 100 : 921.659. Answer. £921 13s. 2d.

EXERCISES IN PER-CENTAge.

(1.) 561 gallons of spirits at 10-8 per cent. O. P. are bought, duty paid, for £36 68. 10d. The price less the duty is 1s. 8d. per proof gallon. How much per cent. does the duty add to the price? Answer. 600 per cent.

(2.) Apples bought at 5s. 9d. the hundred, were sold at a profit equal to 3-7ths. of the selling price. Required the selling price, and the gain per cent.* Answer. 10s. 03d. and 75 per cent. (3.) Leaf tobacco costs 10d. per lb. in bond; the duty is 3s. 2d. If 35 per cent. of water be absorbed during the process of manufacture, what price per lb. will give a profit of 15 per cent. ? Answer. 38. 4 d.

9

(4.) 210 gallons of spirits were bought at 12s. 6d. per gallon. 3 per cent. was lost by leakage, and the remainder was sold at 14s. 2d. a gallon. What was the gain or loss per cent?

exceed?

9.93 gain.

Answer.
Answer. 121 per cent.
What was
Answer. 61s. 11d.

(5.) By how much per cent. does
(6.) By selling malt at 66s. a quarter, 8 per cent. was gained.

the cost price?

(7.) Spirits bought by the bulk gallon at 24.6 per cent. O. P. are reduced to 10 per cent. U. P., and sold at the price per gallon for which they were bought. What is the gain per cent.? Answer. £38 8s. 101d.

* In questions of this kind, the gain per cent. is always calculated on the cost price, unless it be specially mentioned that the per centage is to be found on the selling price.

3

=

2

14

7.

(9.) AVERAGES.-If any set of quantities of the same kind be added together, and their sum be divided by the number of quantities, the quotient, or intermediate value, so obtained, is called the average, or arithmetical mean, of those quantities. Thus, 7 is the average of the numbers 4 and 10, for 4+10 Again, 93 is the average of 4, 10, and 15, since 4+10+15 From the manner in which an average is formed, it follows that the total of its excesses above the quantities which are less than itself, is exactly equal to or balances the total of its defects below the quantities greater than itself. It will be seen for example, on examining the average of the numbers, 1, 2, 5, 9, 13, and 18, that 8, the average in question, when subtracted from 9, 13, and 18 respectively, leaves a series of differences, 1, 5, and 10, the sum of which is 16; and that the numbers 1, 2, and 5, when taken severally from 8, give differences, the sum of which also is 16. Hence, the total of the excesses above the average is the same as the total of the deficiencies below the average; and this property holds good in every instance.

Another view of the nature of an average shows, that it is composed of an equal aliquot part of each of the quantities present, the denominator of such part or fraction being the number of the quantities. Thus, in dividing the sum of 1, 2, 5, 9, 13 and 18 by 6, to find their average, what is done is really equivalent to dividing each of these quantities, by 6, as there are six quantities, and then making a total of the results

A sixth part of each of the quantities being taken to furnish one of the elements of the average, it is plain that the sum of such elements, or the average itself must be intermediate or just midway in point of value, between the different quantities.

(It should be carefully observed, that the average of a set of averages is not the average of the whole, unless the numbers of quantities in each of the sets averaged, are equal. The truth of this will appear on taking the average of the whole without having recourse to the partial averages. If 10 men have on the average £100, and 50 men have on the average £300, the average sum possessed by each person is not the average of £100 and £300, for the 10 men have among them £1,000, and the 50 men have among them £15,000, or £16,000 in all. This divided into 60 parts, gives £266 13s. 4d. to each. A neglect of this remark might lead to erroneous estimates; as, for instance, if a harvest were called good because an average bushel of its corn was better than that of another harvest, without taking into account the number of bushels of the two.

The average quantity is a valuable common-sense test of the goodness or badness of any particular lot or set of quantities, but only when there is a perfect

K

similarity of circumstances in the things compared. For example, no one would think of calling a tree well-grown because it gave more timber than the average of all trees but if any particular tree, say an oak, yielded more wood than the average of all oaks of the same age, that tree would be called good, because if every oak gave as much, the quantity of oak timber generally would be greater than it is. It must also be remembered that the value of the average, in the information which it affords, diminishes as the quantities averaged vary more from one another.*) In all physical inquiries, however, such as the determination of the weights of substances, measures of length or capacity, &c., where a number of values of a quantity have been ascertained with as much precision as possible, by observation or experiment, and where the errors are as likely to be in excess as in defect, the average or arithmetical mean is the most probable result. In the gauging of vessels or solids, and in other matters pertaining to the business of the Excise, it is the invariable practice for the sake of greater correctness, to take an average of two or more measurements of the same dimension or estimates of the same amount, where any sensible differences may exist.

The convenience and propriety of using averages, are well exemplified in the case of malt gauging. See the article under that title in the chapter on « Practical Mensuration.”

To find the average of any number of simple or compound quantities, the RULE is, Add all the quantities together and divide by the number of quantities. Further illustrations cannot be necessary.

There is frequently occasion to compute the average value of a collection or mixture of several quantities, each at a certain rate, price, or other stated value. The method of treating questions of this class is so obvious as hardly to need formal directions.

Add the products

Multiply the quantity of each ingredient by its price or rate. together, and divide the result by the total number of quantities. Example (1.) What is the worth per pound of a mixture of 12 lbs. at 9d., 8 lbs, at 6d., and 5 lbs. at 4d. ?

Here, as is evident, 25 lbs. at the uniform rate of 7.04d. per lb. has the same collective value as when made up of 12 lbs. at 9d., 8 lbs. at 6d., and 5 lbs. at 4d. The 25 lbs. are altogether worth 176d. Consequently, the value of a single pound of the mixture, or the average price per lb. must be 176d. divided by 25.

The process gone through is really one of simple averaging, multiplication of the quantities by their several rates being substituted for the longer task of addition of the number of units of value corresponding to each quantity. Thus, if the question were, what is the worth per gallon of a compound of 4 gallons at 7s., 3 gallons at 6s., and 2 gallons at 4s. ? the work might be performed in two ways, as follows

English Cyclopædia." Arts and Sciences, Article, AVERAGE,

The operation on the right of the vertical line, is only a compendious equivalent of the addition and division on the left.

Example (2.) Suppose a mixture made of 134 gallons of spirits at 51 per cent., O. P.; 218 gallons at 15 per cent, O. P.; 718 gallons at proof; 360 gallons at 8 per cent, U. P.; and 940 gallons at 21 per cent., U. P. What should be the average strength, disregarding the effect of concentration ?

Answer.

Each gallon should have the value of 947 gallon at Proof, or the

whole should be of the strength of 5.3 per cent., U. P.

To obtain by calculation the true average strength of a mixture of spirits of different degrees of strength, it would be necessary to apply a correction to the result calculated as above, for the increase of strength produced by condensation of bulk, but no means of making such a correction exist, nor is the nice adjustment of strength in these cases of any practical importance.

The numbers in the second column of the preceding example are used as multipliers of the bulk quantities, in order to compute the total values at proof, just as rates or prices, per lb., per cwt., &c., would be used in finding the total money value of other quantities.

Example (3.) If 322 gallons of spirits at 22 per cent., O. P., are compounded with 150 gallons at 80 per cent., U. P., and 140 gallons of water, what will be the strength of the mixture?

For the method of finding the values in this column, see Reduction of Spirits, page 98.
The proof equivalent of water is evidently 0.

Example (4.) Required the average gravity of a mixture of 390 gallons of worts at the gravity of 80°, 475 gallons at the gravity of 75°, and 465 gallons of water ?*

Example (5.) If 12 gallons of water be mixed with 55 gallons of spirits worth 12s. per gallon, what will be the value of the mixture per gallon?

Alligation.-Under the title of Alligation, (which signifies a linking or binding together,) most treatises on commercial arithmetic contain a system of rules, chiefly for determining the proportion in which ingredients of given values must be mixed so as to form a compound which shall have a certain intermediate value. The question to be solved in any of these cases-that is, knowing the mean rate and the rates of the ingredients, to determine thence the quantities or proportions of the ingredients-is evidently the reverse of the question which has just been considered, namely, having given the quantities and their rates respectively, to ascertain the mean or average rate of the mixture.

Example (1.) Suppose it asked, In what ratio must two substances, one worth 2s., and the other worth 7d., per lb., be taken, to make a mixture that shall be worth 1s. 4d. per lb. ?

From the general property of an average, as explained at the beginning of this article, (page 137,) it follows in the present instance, that the required quantity or proportion of the ingredient at 2s. multiplied by the excess of that rate above the mean, 1s. 4d., must equal the required quantity of the ingredient at 7d., multiplied by the defect of that rate below the mean.

If the data be all expressed in one denomination as pence, the excess of the higher rate above the mean is 8, and the defect of the lower rate below the mean is 9. We have now to find a number which multiplied by 9, shall give the same product as some other number multiplied by 8. This condition, it is plain, may be satisfied in an infinite variety of ways, provided both the factors are not required to be whole numbers; for whatever number be taken as the multiplier of either difference, an equivalent product will be formed by using the quotient of such product divided by the other difference as a multiplier of that difference. Thus, if 3 be assumed as a multiplier of 9, then

And similarly for every other possible multiplier of 9 or 8. But should the multipliers be restricted to whole numbers, as is generally the case, one very simple mode of answering the question presents itself, and that is, to employ the difference between each rate and the mean alternately, as the multiplier of the other difference.

* For the meaning of the term "gravity" as applied to worts, and the reason of multiplying each quantity by its gravity, see Reduction of Worts, page 104.